TRIANGLE FUZZY TRANSFORM BASED AUTOMATIC NOISE AND COLOR IMAGE REDUCTION METHODS

Noise reduction and image reduction are very important research area for image processing and computer vision. Many papers have been proposed for noise and image reductions. In this paper, novel triangle fuzzy sets transform (F-transform) is proposed for color image denoising and reduction. The proposed methods consist of histogram extraction, threshold points calculation, fuzzy sets construction and fuzzy tansformation phases. Firstly, histogram of the image is extracted, maximum points of histogram are calculated, and these points are considered as threshold points. Fuzzy sets are created using threshold points. Then, F-transform is applied on the overlapping and non-overlapping blocks of the images for image denoising and reduction respectively. The main objective of the presented method are to remove random noises of the images and color image reduction with satisfactory visual quality. In order to evaluate triangle fuzzy sets based Ftransform applications, variable noise intensities and block sizes are used. Mean absolute error (MAE), peaks signal noise-to-ratio (PSNR) and penalized function (PEN) are utilized for obtaining numerical results. Numerical simulations and comprasions clearly illustare that the proposed triangle F-transform is good transformation for random noises removing and image reduction.


Introduction
In the literature, many transformations for instance Discrete Wavelet Transform (DWT), Discrete Cosine Transform (DCT), Fractional Fourier Transform (FrFT), Fast Fourier Transform (FFT) have been proposed for image reduction, compression, watermarking, recognition and classification [1][2][3][4]. Fuzzy transform (F-transform) is one of them and it was presented by Perfilieva in the 2006 [5]. Ftransform have been used very large area in the image processing such as image reduction, compression, Step 3: Divide histogram into N parts. N>1. N represents number of the fuzzy sets.
Step 4: Calculate threshold points using Algorithm 2. Algorithm 2. Pseudo code of the threshold points calculation.
Input: Histogram of the image (histo) with size of 256.
Output: Threshold points (thr) with size of N.

5: endfor
Step 5: Calculate membership degrees of the images using threshold points.
Where ℎ , pixel of the channel, ℎ is threshold point, i and j are indices of the image, W is width of the image (channel), H represents height of the image and k is indice of the threshold points and = {1,2, … , }. Eq. 1 defines triangle fuzzy sets and a sample of it is shown in Fig. 1.  Step 6: Divide image into b x b size of overlapping blocks using noise intensity.
Step 7: Apply F-tranform to each overlapping block.
Step 8: Combine the R, G and B channels and reconstruct fuzzy image.

The proposed F-Transform based image reduction method
In here, triangle fuzzy transform based color image reduction is presented. This method is similar to Martino et al.'s [6] method. In Ref. [6], cosine based fuzzy sets are used. In this paper, triangle based fuzzy sets are used to reduct color images. The pseudo code of the presented color image reduction algorithm is given below.

Experimental Results
In this section, numerical results of the proposed methods are obtained for performance evaluation and PSNR, MAE and Penalty function (PEN) metrics are considered to get numerical results from test images. Mathematical description of the PSNR, MAE and PEN are given Eq. 5-7. Also, 2 triangle fuzzy sets were used to obtain experiments because the best results are achieved using 2 triangle fuzzy sets.
= 10 log 10 where is original image and reduced image for noise or image reduction.
In order to achieve numerical results, a color image test set is used. The test images used are shown in Fig. 2. Visual results of samples were shown in Fig. 3 to better understand success of the proposed noise reduction method.  Table 2. In this section, 2 x 2, 3 x 3, 4 x 4 and 5 x 5 size of blocks are used to image reduction and one pixel is constructed from each block. To compare this method to others, PEN values of methods with 3 x 3 size of blocks are used and comparisons are listed in Table 3.

Discussions
Discussions of this study are given as below.
 The proposed noise reduction algorithm is simple as median and mean filters. It achieved very successful results for random noises and Table 1, Fig. 2 proved that this success.
 Results of the proposed image reduction method was listed in the Table 2 and comparisons were given in the Table 3. This method was compared to 3 state of art methods using PEN values and 10 colored test images. The average PEN value of the proposed method was calculated as 475.26. It is the best value among them. Table 3 clearly demonstrated that the proposed method also achieved the best PEN values for 5 test images.

Conclusions and recommandations
In this study, a novel triangle based F-transform is proposed and its applications, which are image denoising and reduction, are presented with two novel methods. In the noise reduction method non-overlapping blocks are used. This method uses no correction steps and succesful results are achived using the proposed triangle F-transform based noise reduction method.
Random noises were used to implement simulations with 10-80% noise intensities. The proposed denoising method is a basic method and it is better than mean and median filters for random noises. By using non-overlapping blocks, a novel image reduction method is presented and the comparison results have been clearly demonstrated that the proposed image reduction method is a good image decomposition method.
In the future studies, novel applications for instance perceptual hash, image watermarking, etc. methods will be proposed and novel F-transforms will be proposed using other fuzzy sets.